Best Simultaneous Approximation in L p (I,X)

Transcription

1 nt. Journal of Math. Analysis, Vol. 3, 2009, no. 24, Best Simultaneous Aroximation in L (,X) E. Abu-Sirhan Deartment of Mathematics, Tafila Technical University Tafila, Jordan Esarhan@ttu.edu.jo Abstract Let X be a Banach Sace and let G be a closed subsace of X. We say G is simultaneously roximinal in X if for any x,x 2 in X, { there exists some y G such } thatd ({x, x 2 },G)( x y + x 2 y ) inf ( x z + y z ) : z G. n this aer we give an inequality involving d ({x, x 2 },G) in vector valued integrable functions. Results on simultaneous roximanlity in such saces will be resented. Keywords: Simultaneous Aroximation, Distance nequality ntroduction Let G be a closed subsace of a { Banach sace X. For a finite subset E X and ( ) }, we write: d (E,G) inf Σ e y e E : y G. Such infimum need not be attained. n case the infimum is attained for any subset E X, we say that G is E Simultaneously roximinal under the norm, or simly ( E,) simultaneously roximinal; where E is the cardinality of E. n the case E, (, ) simultaneous roximinality is just roximinality. C.B. Dunham, J.B. Diaz and H.W. McLaughlin (see[2],[5]) have considered the roblem of best simultaneous aroximation in the following case: X C[a, b], the sace of continuous real valued functions on some comact interval [a, b], Gis a non emty subset of X and E {f,f 2 }, d (E,G) inf {max( g f, g f 2 ):g G} which is (2, max) simultaneously roximinality roblem.

2 58 E. Abu-Sirhan Goel, Holand, Nasim, and Sahney [7] studied the roblem when X is a normed linear sace, G is any subset of X, and F {x,x 2 } X. Many good result had aeared since then. We refer to [4], [], [0], [8], and [6]. However, all these results deal with the sace of continuous functions with d. We let L (,X) be the Banach sace of all Bochner integrable functions defined on with values on X; where [0, ] is the unit interval in set of real numbers R. Here R is the set of real numbers. The norm of f L (,X) is given by f ( f dμ / ) ; where μ is the Lebesgue measure on. t is the object of this aer to study (2,P) simultaneous aroximation in L (,X). This roblem has been studied by several authors ( see [3], [9], []). n [], the roblem has been studied with d forl (,X). n [3] and [9] the roblem has been studied for some classes of G. 2 L summand and simultaneous roximinality in L (,X) Lemma 2. let X be a Banach sace and x, y X, then x + y 2 ( x + x ), for any natural number. roof. The roof uses mathematical induction. The case is just the triangle inequality, so the assertion is valid in this case. Next, assume the validity of the inequality for k, that is x + y k 2 ( x k k + y k). Now we will use the Arithmetic Geometric Mean nequality to rove the validity for k +. x + y k+ x + y k x + y 2 k ( x k + y k) x + y ( 2 k x k + y k) ( x + y ) ( ) 2 k x k+ + y k+ + x k y + y k x ( 2 k x k+ + y k+ + k + (k +) x k y + 2 k ( x k+ + y k+ + k + ( k x k+ + y k+) + k + ) k + (k +) y k x (k y k+ + x k+)) 2 k ( 2 x k+ +2 x k+) 2 k ( x k+ + x k+). Ξ

3 Best simultaneous aroximation in L (,X) 59 Definition 2.2 Let X be a Banach sace and G be a closed subsace of X. Then G is called L summand of X, if X G Ĝ such that x + y x + y for any x G and y Ĝ. Lemma 2.3 Let X be a Banach sace and G be an L summand of X. Then G is (2,) simultaneously roximinal in X. Proof. Let x, y X G Ĝ, x x + x 2 and y y + y 2. We will show that θ x +y 2 2 is a best simultaneous aroximation for x, y from G. x θ x + x 2 x + y 2 2 x 2 + ( ) y x 2 Then, we have y θ y 2 + ( ) y x 2 x θ + y θ x 2 + y 2 +(2) y x. Let z G, using lemma 2. x θ + y θ x 2 + y 2 +(2) y z + z x x 2 + y 2 +(2) ( 2 ( x z + y z ) ) x 2 + y 2 + x z + y z x 2 + x z + y 2 + y z x z + y z. Since z was arbitrary, then ( x θ + y θ ) ( x z + y z ), for any z G and d ({x,x 2 },G)( x θ + y θ ). Ξ

4 60 E. Abu-Sirhan Let z X Y, where X and Y are Banach saces. For, the nuclear norm { of z is defined by the following equations, in which +q and μ q (y,..., y n ) su n λ i y i :( } n λ i ) / : ( ) / α (z) inf x i μ q (y,..., y n ):z x i y i. The infimum is taken with resect to all reresentations of z. We denote the comletion of X Y with resect to the α by X Y. From the theory of tensor roduct (see [2] α() ), it is known that L (,X)L () X, for any Banach sace X. α() Theorem 2.4 Let X be a Banach sace and G be an L summand of X. Then L (,G) is (2,) simultaneously roximinal in L (,X). Proof. By lemma 2.3, it is sufficient to show that L () α() G is an L summand of L () X. Let P : X G be the rojection and id be the identity function on L (). α() Then id : L () X L () G, α() α() defined by id P (f x) f P (x). Clearly id is a bounded linear rojection. Now let f L (,X)L () X be a simle function, so that f n Ei x i.we α() may assume that E i : i, 2,..., n, are air wise disjoint and n Ei ; where is the constant function. f Ei P (x i )+ Ei (x i P (x i )) f Ei (t) x i dμ E i Ei (t) x i dμ E i Ei (t) x i dμ

5 Best simultaneous aroximation in L (,X) 6 μ(e i ) x i μ(e i )( P (x i ) + x i P (x i ) ) μ(e i ) P (x i ) + μ(e i ) x i P (x i ) Ei P (x i ) + Ei (x i P (x i )) (id P )(f) + f (id P )(f). Since simle functions are dense in L (,X), it follows that f (id P )(f) + f (id P )(f), f L (,X). Thus L () G is an L summand of L () X. Ξ α() α() 3 The Distance nequality n this section we deduce a distance inequality for best simultaneous aroximation in L (,X).For any Banach sace Y and a closed subsace G of X, and, we set Y Y with (x, y) ( x + y ),and D(G) {(g,g) :g G} with (g,g) ( g + g ).t is clear that D(G) is a closed subsace of the Banach sace Y Y. Theorem 3. Let X be a Banach sace,f,f 2 L (,X), and G be a closed subsace of X. Define φ(t) d ((f (t),f 2 (t)), D(G)). Then φ L () and ( ) φ(t) dt d ((f, f 2 ),D( L (,X)) ) 2 ( ) φ(t) dt. () Proof. Let f,f 2 L (,X). The there exist two sequences (f n ), (f 2n ) of simle functions such that lim f (t) f n (t) 0 a.e. and lim f 2 (t) f 2n (t) 0 a.e.

6 62 E. Abu-Sirhan Since the distance is a continuous function, then lim d ((f n (t), f 2n (t)), D(G) )d ((f (t), f 2 (t)), D(G) )φ(t) a.e. We may write f n (t) n Ei x i and f 2n (t) n Ei y i. We may assume that E i : i, 2,..., n, are air wise ()disjoint and Ei ; where is the constant { } function. Then, we set φ n (t) inf ( f n (t) z + f 2n (t) z ) : z G, n { } φ n (t) inf ( f n (t) z + f 2n (t) z ) : z G ( ) inf Ei (t) x i z + Ei (t) y i z { } inf Ei (t)( x i z + y i z ) : z G Ei (t) inf { ( x i z + y i z ) Ei (t) d ((x i, y i ), D(G) ). } : z G : z G Hence φ n is a simle function for all n. Consequently φ is measurable. Let g L (,G),then ( f g + f 2 g ) f (t) g (t) dt + f 2 (t) g (t) dt ( f (t) g (t) + f 2 (t) g (t) )dt d ((f (t),f 2 (t)), D(G)) dt, take the inf over all g L (,G), we get the st inequality. To rove the second inequality, let ɛ>0 and f,f 2 be simle functions in L (,X) such that f f < ɛ and f (4)2 2 f 2 < ɛ. We may write f (4)2 (t) n Ei (t) x i and f 2 (t) Ei (t) y i. Also we may assume that E i : i, 2,..., n, are air wise disjoint, Ei

7 Best simultaneous aroximation in L (,X) 63 ; where is the constant function, and μ (E i ) > 0 for i, 2,..., n. Now, for each i, let h i G be such that x i h i + y i h i < d ((x i,y i ),D(G)) + ɛ 4 nμ(e i )... (). Let g n Ei h i, then g L (,G). Then d ((f,f 2 ), D(L (,G))) inf { } ( f h + f 2 h ) : h L (,G) ( f g + f 2 g ) ( f f + f g f2 + f 2 + f 2 g ) ( f f + f 2 f 2 ) ( f + g + f 2 g ) (( ) ɛ ( < (4) 2 / + ɛ (4) 2 / ) ) + ( f g + f 2 g ) ɛ ( f 4 + g + f 2 g ) ɛ 4 + ( μ (E i )( x i h i + y i h i ) ). Using inequality (), we get ( d ((f,f 2 ), D(L (,G))) ɛ ( 4 + (E i ) d ((x i,y i ),D(G)) μ + ɛ ) ) 4 nμ(e i ) ɛ 4 + ( μ (E i ) d ((x i,y i ), D(G)) + ɛ 4 )

8 64 E. Abu-Sirhan ɛ 4 + ( μ (E i ) d ((x i,y i ),D(G)) ) + ɛ 4 ɛ 2 + ) d ((f (t),f 2 (t) by using lemma 2., we get d ((f,f 2 ), D(L (,G))) ɛ 2 + ),D(G) dt { f inf (t)+f (t) f (t)+h + f 2 (t)+f 2 (t) f 2 (t)+h } : h G dt ɛ 2 + ( 2 f inf (t)+f (t) + f (t) h ) + ( 2 f 2 (t)+f 2 (t) + f 2 (t) h ) : h G dt ɛ ( f 2 (t) h + f (t) h )+ inf ( 2 f 2 (t)+f 2 (t) + f (t)+f (t) ) : h G dt ɛ 2 + inf { 2 ( f 2 (t) h + f (t) h ):h G } dt+ ( f 2 (t)+f 2 (t) + f (t)+f (t) ) dt 2 ɛ 2 + inf { 2 ( f 2 (t) h + f (t) h ):h G } ( ) ɛ dt +2 4 (2) + ɛ 4 (2) ɛ 2 + inf { 2 ( f 2 (t) h + f (t) h ):h G } dt +2 ( ɛ 4 ) ɛ 2 +2 inf {( f 2 (t) h + f (t) h ):h G} dt + ɛ 2

9 Best simultaneous aroximation in L (,X) 65 ɛ +2 d ((f (t),f 2 (t)),d(g)) dt ɛ +2 ( φ(t)) dt Since ɛ was arbitrary, we get the second inequality. Ξ The following corollaries show when the equality hold for the inequality of theorem 3.. The first corollary was the main result in [].. Corollary 3.2 Let X be a Banach sace,f,f 2 L (,X), and G be a closed subsace of X. Define φ(t) d ((f (t),f 2 (t)), D(G)). Then φ L () and φ(t) dt d ((f, f 2 ), L (,G)). Proof. Take in Theorem 3.. Ξ Corollary 3.3 Let X be a Banach sace,f,f 2 L (,X) be simle functions, and G be a closed subsace of X.Then ( ) d ((f (t),f 2 (t)), D(G)) dt d ((f, f 2 ),L (,G)). Proof : By theorem 3., it is sufficient to show that d ((f, f 2 ), L (,G) ) ( d ((f (t),f 2 (t)), D(G)) dt ). Let ɛ > 0, and use the same techniques used in the roof of Theorem 3. for the simle functions f,f 2, we get d ((f, f 2 ), L (,G) ) ɛ 2 + ( d ((f (t),f 2 (t)), D(G)) dt ).Since ɛ was arbitrary, the inequality is roved. Ξ Theorem 3.4 Let X be a Banach sace and Gbe a closed subsace. Then for a strongly measurable function g: G we have :. For f, f 2 L (,G), if g (t) is a best simultaneous aroximation of f (t),f 2 (t) a.e. in X from G, then g is a best simutaneous aroximation of f, f 2 from L (,G). 2. f g is a best simultaneous aroximatant of simle functions f,f 2 L (,X) from L (,G), then g (t) is a best simultaneous aroximation of f (t),f 2 (t) a.e. in X from G.

10 66 E. Abu-Sirhan Proof. () : Assume that g (t) is a best simultaneous aroximation of f (t),f 2 (t) a.e. in X from G, then ( f (t) g (t) + f 2 (t) g (t) ) ( f (t) z + f (t) z ) a.e. and for any z G. Using triangle inequality and take z 0, ( g (t) + g (t) ) 2( f (t) + f (t) ) a.e., g (t) 2 ( f (t) + f (t) ), hence g L (,G).Now, let h L (,G), then ( f (t) g (t) + f 2 (t) g (t) ) ( f (t) h (t) + f (t) h (t) ) a.e., f g + f 2 g f h + f h. Since h L (,G) was arbitrary, then ( f g + f 2 g ) d ((f, f 2 ),D(L (,G)) ). (2) : Assume that g is a best simultaneous aroximation of simle functions f,f 2 L (,X) from L (,G). Using Corollary 3.3, ( f g + f 2 g ) ( f (t) g (t) + f 2 (t) g (t) )dt d ((f (t),f 2 (t)),d(g) ) dt d ((f (t),f 2 (t)), D(G) ) dt., Since f (t) g (t) + f 2 (t) g (t) d ((f (t),f 2 (t)), D(G) ) ( f (t) g (t) + f 2 (t) g (t) ) d ((f (t),f 2 (t)), D(G) ) a.e. Ξ a.e., then Theorem 3.5 Let X be a Banach sace and G be a closed subsace of X. f L (,G) is simultaneously roximinal in L (,X), then G is simultaneously roximinal in X. Proof : Let x, y X. Set f x x, f y y ; where is the constant function on. Since (,μ) is a finite measure sace, then L (), hence f x, f y L (,X).Since

11 Best simultaneous aroximation in L (,X) 67 L (,G) is simultaneously roximinal in L (,X), then there exists g L (,G) such that ( f x g + f y g ) d ((f, f 2 ), D(L (,G)) ). Using Theorem 3.4, f x (t) g (t) + f y (t) g (t) d ((f x (t),f y (t)),d(g) ) a.e., ( x g (t) + y g (t) ) d ((x, y),d(g) ) a.e.. n articular ( x g (t 0 ) + y g (t 0 ) ) d ((x, y), D(G) ) for some t 0. Thus g (t 0 ) is a best simultaneous aroximation of x, y in X from G. Ξ ACKNOWLEDGEMENTS. should like to thank Pro. R. Khalil for his valuable advice. References [] Abu-Sirhan, R. Khalil, Best Simultaneous Aroximation in Function Saces, nt.journal of Math. Analysis, vol.2 (2008), no. 5, [2] A. S. B. Holand, B. N. Sahney and J. Tzimbalario, On Best Simultaneous Aroximation, J. ndian Math. Soc. 40(976), [3] B. Fathi, D. Hussein, R. Khalil, Best Simultaneous Aroximation in L P (,E), J. Aroximation Theory 6(2002), [4] B. N. Sahney and S. P. Singh, On Best Simultaneous Aroximation in Banach Saces, J. Aroximation Theory 35(982), [5] C. B. Dunham, Simultaneous Chebychev Aroximation of Functions on an nterval, Proc. Amer. Math. Soc. 8(976), [6] Chong Li, On Best Simultaneous Aroximation, J. Aroximation Theory 9(997), [7] D. S. Goel, A. S. B. Holand, C. Nasim, and B. N. Sahney, On Best Simultaneous Aroximation in Normed Saces, Canad. Math. Bull. 7 (974), [8] G. A. Watson, A Characterization of Best Simultaneous Aroximation, J. Aroximation Theory 75(993),

12 68 E. Abu-Sirhan [9] J. Mendoza, T. Pakhrou, Best Simultaneous Aroximation in L (,E),J. Aroximation Theory 45(2007), [0] Shinji Tanimoto, A Characterization of Best Simultaneous Aroximation, J. Aroximation Theory 59(989), [] Shinji Tanimoto, On Best Simultaneous Aroximation, Math. Jaonica 48, No 2(998), [2] W.A.Light, E.W. Cheney, Aroximation Theory in Tensor Product saces, Lecture Note in Mathematics 69, 985. Received: December, 2008